A272283 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 457", based on the 5-celled von Neumann neighborhood.

3, 5, 19, -7, 71, -35, 111, -103, 179, -163, 271, -255, 439, -371, 491, -423, 659, -671, 903, -847, 1155, -1071, 1315, -1251, 1499, -1483, 1907, -1763, 2031, -1839, 2063, -1935, 2275, -2403, 3011, -2855, 3279, -3167, 3559, -3431, 3835, -3783, 4415, -4047

Offset

0,1

Comments

Initialized with a single black (ON) cell at stage zero.

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Robert Price, Table of n, a(n) for n = 0..127

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=457; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

Crossrefs

Cf. A272280.

Sequence in context: A270894 A272317 A128362 * A053484 A272092 A272019

Adjacent sequences: A272280 A272281 A272282 * A272284 A272285 A272286

Keyword

sign,easy

Author

Robert Price, Apr 24 2016

Status

approved